Proposition 18

Let $A$ be a ring and $S$ a multiplicative subset of $A$.

(i) If $M$ and $N$ are two $A$-modules, the $S^{-1}A$-modules $(S^{-1}M) \otimes_A N, M \otimes_A (S^{-1}N)$, $(S^{-1}M) \otimes_{S^{-1}A} (S^{-1}N)$ and $S^{-1}(M \otimes_A N)$ are canonically isomorphic.

(ii) If $M$ and $N'$ are two $S^{-1}A$-modules, the canonical homomorphism

derived from the $A$-bilinear mapping $(x', y') \mapsto x' \otimes y'$ from $M' \times N'$ to $M' \otimes_{S^{-1}A} N'$ is bijective Assertion (i) is an immediate consequence of the definition $S^{-1}M =$

M \otimes_A S^{-1}A and the associativity of tensor products, which gives to within canonical isomorphisms

To prove (ii), we note first that in $M'$ and $N'$, considered as $A$-modules, the homotheties induced by the elements $s \in S$ are bijective, hence $M' = S^{-1}M'$ and $N' = S^{-1}N'$ (no. 2, Remark 4) and similarly $S^{-1}(M' \otimes_A N') = M' \otimes_A N'$; (ii) is then a special case of (i).